Pilot-Wave Simulator: Exact Classical Sampling from Ideal and Noisy Quantum Circuits up to Hundreds of Qubits

TL;DR

The authors address the challenge of exactly sampling from quantum circuits on classical hardware, especially for structured, non-random circuits like QAOA, by introducing the Pilot-Wave (PW) simulator which couples a Markov-chain sampling process with a tensor-network amplitude oracle. The method supports realistic noise models through a noise-augmented tensor-network framework and a gate-propagation strategy that preserves contraction complexity, enabling exact sampling up to hundreds of qubits. Key findings include the ability to produce large-scale ideal and noisy QAOA samples, observation of pseudo-Boltzmann distributions with depth, and a quantitative comparison to Hastings’ classical local-update algorithm showing regimes where classical methods are competitive or superior. The work provides practical baselines for near-term quantum hardware and informs the design and benchmarking of quantum algorithms on NISQ devices, with broad implications for classical verification and algorithmic development in quantum computing.

Abstract

Quantum circuit simulators running on classical computers offer a vital platform for designing, testing, and optimizing quantum algorithms, driving innovation despite limited access to real quantum hardware. However, their scalability is inherently constrained by exponential memory and computational overhead, which restricts accurate simulation of large-scale quantum circuits and often results in approximate output distributions. Here, we propose an exact sampling algorithm that integrates tensor network contraction techniques with a Markov process, wherein a classical state evolves according to the local structure of the quantum circuit. As a demonstration, we target the challenge of generating samples from ideal and noisy QAOA circuits with up to 476 qubits, incorporating both depolarizing and amplitude damping noise models. These results enable further validation of several assumptions and conjectures at a scale previously out of reach, significantly expanding the scope of classical simulation in quantum algorithm research.

Figures (5)

• Figure 1: Panel (a) shows that $U_t$ has the block structure of $A \otimes I_{2^{n-m}}$ (up to a basis permutation). Each block corresponds to the gate matrix $A$, which can be either (b) a monomial gate (i.e., exactly one nonzero per row and column) or (c) a general gate. Matrices of general gates are further partitioned into blocks using the BlockDecompose procedure from Algorithm \ref{['al:sample']}.
• Figure 2: Sampling results for QAOA circuits obtained with a simulated quantum device. Panels (a,b) correspond to the ideal device; panels (c,d) to the noisy chip. (a,c) Normalized histograms of measurement outcomes from $1.2 \cdot 10^8$ samples on $N=36$ qubits arranged on a rectangular grid. (b,d) Scaling with system size $N$ of (i) the ground state probability and (ii) the probability of the empirically most frequent bitstring in the sampled batch. The dashed black line shows the uniform baseline $2^{-N}$.
• Figure 3: Experimental results of sampling from QAOA circuit in comparison to classical Hastings algorithm. Each count on the diagram corresponds to the lowest energy out of 100 samples from the corresponding algorithm.
• Figure 4: Sampling times of QAOA circuits with $p=1$, averaged over $10$ independent problem instances for different graph topologies and problem sizes.
• Figure 5: Fusion and propagation of an $X$ gate in a quantum circuit.

Theorems & Definitions (2)

• Proposition 1
• Proposition 2